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Value sets of bivariate folding polynomials over finite fields
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Date
2018-11-01
Author
Küçüksakallı, Ömer
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We find the cardinality of the value sets of polynomial maps associated with simple complex Lie algebras B-2 and G(2) over finite fields. We achieve this by using a characterization of their fixed points in terms of sums of roots of unity.
Subject Keywords
Lie algebra
,
Weyl group
,
Fixed point
,
Permutation
URI
https://hdl.handle.net/11511/32985
Journal
FINITE FIELDS AND THEIR APPLICATIONS
DOI
https://doi.org/10.1016/j.ffa.2018.05.008
Collections
Department of Mathematics, Article
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Value sets of folding polynomials over finite fields
Küçüksakallı, Ömer (2019-01-01)
Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Bivariate polynomial mappings associated with simple complex Lie algebras
Küçüksakallı, Ömer (2016-11-01)
There are three families of bivariate polynomial maps associated with the rank-2 simple complex Lie algebras A(2), B-2 congruent to C-2 and G(2). It is known that the bivariate polynomial map associated with A(2) induces a permutation of F-q(2) if and only if gcd(k, q(3) - 1) = I. for s = 1, 2, 3. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.
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Exceptional Lie algebra g2 and its representations
Kayakökü, Mehmet Mustafa; Ünal, İbrahim; Department of Mathematics (2022-9-01)
In the classification of complex simple Lie algebras, there are five of them whose Dynkin diagrams are of exceptional type. The Lie algebra g_2 has the smallest dimension among these exceptional Lie algebras and together with its corresponding Lie group G_2, it plays an important role in differential geometry, mathematical physics, and modern string theory. In this thesis after a general introduction to Lie algebras, we show the classification of complex simple ones. Afterward, we give several constructions...
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Ö. Küçüksakallı, “Value sets of bivariate folding polynomials over finite fields,”
FINITE FIELDS AND THEIR APPLICATIONS
, pp. 253–272, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32985.